Name: ___________________

Math 122 Lab 4: Calculator Integration

How Good is the Calculator?

As you get further into calculus, you will rely more on the calculator to supply complicated integrations. Before you place too much trust in the machine, you should explore the reliability of its integration routine. The bottom line is that there are a few minor and fairly predictable flaws in the calculator that you should be aware of. Otherwise, the TI-89 is very good!

What to Turn In

Turn in this worksheet with everything filled in. This will be graded for accuracy and understanding, so write good explanations when you are asked to explain something. In addition, turn in a short (about one page, three or four paragraphs) paper summarizing what you have learned. In particular, what do you need to be aware of when you are using the calculator to evaluate an integral? Indicate which calculator ability you were most impressed with, and indicate which calculator mistake you found most disturbing or surprising. This is due Friday at the beginning of class. Papers turned in later Friday lose 1 point, papers turned in Monday or later lose 4 points per day late.

Basic Integrals

Start by asking the calculator for ∫ 1/x dx. Write its answer here and explain what minor mistake the calculator makes.

Now try ∫ cos x / (sinx−2) dx. Note that the TI-89 does supply the absolute value sign. Explain why this answer would be wrong without the absolute value signs.

So, we found one (minor) mistake and one impressive feature (absolute values are included). Most problems that you have with the calculator will be syntax errors (in other words, your fault). The TI-89 allows variable names to be more than one letter long. Try the following integral exactly as written

∫ (xsin(x),x)

Write down the calculator’s response here.

Now, edit the command to put a multiplication * in between the x and sinx. Write down the calculator’s response here.

Finally, try ∫ (xsinx,xsinx) and explain what the calculator thinks the difference is between xsinx and x*sin(x).

Try the following integral exactly as written.

∫ (4x8x,x)

Write the calculator’s response here and explain what it thought it was doing. (Hint: this is a variation on the previous problem.)

Another version of this error is in the command ∫ (x(3x+2),x) . Write the calculator’s response here.

Recall that if you have a function stored in the graphing list as y1, you can plug in x = 2 by using the command y1(2). Given this reminder, what is there about the command x(3x+2) that could confuse the calculator? Discuss this and indicate how to correctly type in the integral ∫ x(3x+2) dx.

More Complicated Integrals

In Chapter 7, you are studying specialized techniques for evaluating certain types of integrals. In the next few steps, you can find out how much of this the calculator knows. How about integration by parts? Try ∫ x⁴ sin2x dx. Write down the calculator’s answer and discuss any ways in which it differs from the answer you would have found working by hand.

Next, try one of the hard integration by parts problems, where you had to do a substitution and integrate by parts. Try ∫ sin dx. Write down the calculator’s answer and discuss any ways in which it differs from the answer you would have found working by hand.

Next, partial fractions. Try ∫ (x²-3) / (x³−x) dx. Write down the calculator’s answer and discuss any ways in which it differs from the answer you would have found working by hand.

Impressive so far, but let’s get tougher. It should be able to do ∫ 1 / [x²(3+2x)] dx. Check and make sure. Write down its answer. (Be careful to get the correct syntax on this one.)

Now, if you let x = sinx, the integral becomes ∫ (cosx) / [(sin²x)(3+2sinx)] dx. Try this ugly one. Write down the calculator’s response.

Based on this, do you think the calculator actually does substitution? Briefly discuss.

There actually are some integrals that the calculator can’t do. Try ∫ dx. Write down the calculator’s answer and explain what it means.

There are other versions of this response. Try ∫ dx. Write down the calculator’s answer and explain what it means. What do you think the calculator did to get its answer?

One other source of confusion is common in trig integrals. Try ∫ dx. Write down the calculator’s answer. If the back of the book shows a correct answer of + c, discuss the accuracy of the TI-89’s answer.

Two Explorations

The calculator can let us try out some integrals and look for patterns. In each case, write down the calculator’s value of each listed integral and look for patterns. You won’t be able to give an exact answer, but say as much as you can about the value of the general integral.

∫ dx :

∫ dx :

∫ dx :

Guesses for general n, ∫ dx :

dx :

dx :

dx :

Guesses for general n, dx :